While Shatranj is considered by most the mother of chess, would it be considered a chess variant? Or is it another board game?
Fairy chess or hexagonal chess are both considered to be chess variants, and are maybe further away from chess than Shatranj.
If anything, it's chess itself which should be called a shatranj variant!
Now, I'm not sure whether to consider a variant is relevant (is this a tag issue question?), but yes, I think shatranj would be considered a chess variant, since the main rule (the checkmate idea) was already in place.
There may be similarities (8x8 board, similar starting position) and differences (no castling, no draw with stalemate, different moves for several pieces), but the main spirit of chess remains the same and I think that's why chess variants are all about.